To solve for the marginal profit at [tex]\( q = 2 \)[/tex] using the given data, we follow these steps:
1. Understand the Marginal Profit Formula:
The marginal profit at a certain quantity [tex]\( q \)[/tex] is defined as the change in revenue minus the change in cost when producing one additional unit. Mathematically, if [tex]\( P(q) \)[/tex] represents profit at quantity [tex]\( q \)[/tex], then the marginal profit at [tex]\( q \)[/tex] can be calculated as:
[tex]\[
MP(q) = \Delta R(q) - \Delta C(q)
\][/tex]
where [tex]\( \Delta R(q) \)[/tex] is the change in revenue and [tex]\( \Delta C(q) \)[/tex] is the change in cost:
[tex]\[
\Delta R(q) = R(q+1) - R(q)
\][/tex]
[tex]\[
\Delta C(q) = C(q+1) - C(q)
\][/tex]
2. Identify [tex]\( q = 2 \)[/tex] in the table:
At [tex]\( q = 2 \)[/tex]:
[tex]\[
R(q) = 10 \quad \text{and} \quad R(q + 1) = 15
\][/tex]
[tex]\[
C(q) = 12 \quad \text{and} \quad C(q + 1) = 15
\][/tex]
3. Calculate the changes in revenue and cost:
[tex]\[
\Delta R(2) = R(3) - R(2) = 15 - 10 = 5
\][/tex]
[tex]\[
\Delta C(2) = C(3) - C(2) = 15 - 12 = 3
\][/tex]
4. Compute the marginal profit:
[tex]\[
MP(2) = \Delta R(2) - \Delta C(2) = 5 - 3 = 2
\][/tex]
Thus, the marginal profit at [tex]\( q = 2 \)[/tex] is [tex]\( 2 \)[/tex] dollars.
